Up \( \partial_1 \) の表現行列 作成: 2023-10-11
更新: 2023-10-11

    \( C_1 \) の基底を，つぎのように定めた：

    \( C_0 \) の基底は，\( \{ v_1, v_2, v_3, v_4 \} \) をとる。

    各 \( e_i \) のバウンダリは，
      \[ \begin{align} \partial e_1 = v_2 - v_1 = &- v_1 &+ v_2 & & \\ \partial e_2 = v_3 - v_1 = &- v_1 & &+ v_3 & \\ \partial e_3 = v_4 - v_1 = &- v_1 & & &+ v_4 \\ \partial e_4 = v_3 - v_2 = & &- v_2 &+ v_3 & \\ \partial e_5 = v_4 - v_2 = & &- v_2 & &+ v_4 \\ \partial e_6 = v_4 - v_3 = & & &- v_3 &+ v_1 \\ \end{align} \]

    よって，基底 \( \{ e_i \}, \{ v_j \} \) に対する \( \partial_1 : C_1 \rightarrow C_0 \) の表現行列は，
      \[ \begin{align} \begin{array}{c c} & \begin{array} {@{} c c c c @{}} v_1 & v_2 & v_3 & v_4 \end{array} \\ \begin{array}{c} e_1 \\ e_2 \\ e_3 \\ e_4 \\ e_5 \\ e_6 \end{array} \hspace{-1em} & \left( \begin{array}{@{} c c c c @{}} -1 & 1 & & \\ -1 & & 1 & \\ -1 & & & 1 \\ & -1 & 1 & \\ & -1 & & 1 \\ & & -1 & 1 \\ \end{array} \right) \\ \mbox{} % Blank line to match column names so as to align the = vertically \end{array} \\[-12pt] % Correction for blank line \end{align} \]

    これに対し，

    \( e'_5 = e_5 - e_3 \)
    \( e'_6 = e_6 - e_3 \)
      \[ \begin{align} \begin{array}{c c} & \begin{array} {@{} c c c c @{}} v_1 & v_2 & v_3 & v_4 \end{array} \\ \begin{array}{c} e_1 \\ e_2 \\ e_3 \\ e_4 \\ e'_5 \\ e'_6 \end{array} \hspace{-1em} & \left( \begin{array}{@{} c c c c @{}} -1 & 1 & & \\ -1 & & 1 & \\ -1 & & & 1 \\ & -1 & 1 & \\ 1 & -1 & & \\ 1 & & -1 & \\ \end{array} \right) \\ \mbox{} % Blank line to match column names so as to align the = vertically \end{array} \\[-12pt] % Correction for blank line \end{align} \]

    \( e'_4 = e_4 - e_2 \)
    \( e''_5 = e'_5 + e_1 \)
    \( e''_6 = e'_6 + e_2 \)
      \[ \begin{align} \begin{array}{c c} & \begin{array} {@{} c c c c @{}} v_1 & v_2 & v_3 & v_4 \end{array} \\ \begin{array}{c} e_1 \\ e_2 \\ e_3 \\ e'_4 \\ e''_5 \\ e''_6 \end{array} \hspace{-1em} & \left( \begin{array}{@{} c c c c @{}} -1 & 1 & & \\ -1 & & 1 & \\ -1 & & & 1 \\ 1 & -1 & & \\ & & & \\ & & & \\ \end{array} \right) \\ \mbox{} % Blank line to match column names so as to align the = vertically \end{array} \\[-12pt] % Correction for blank line \end{align} \]

    \( e''_4 = e'_4 + e_1 \)
      \[ \begin{align} \begin{array}{c c} & \begin{array} {@{} c c c c @{}} v_1 & v_2 & v_3 & v_4 \end{array} \\ \begin{array}{c} e_1 \\ e_2 \\ e_3 \\ e''_4 \\ e''_5 \\ e''_6 \end{array} \hspace{-1em} & \left( \begin{array}{@{} c c c c @{}} -1 & 1 & & \\ -1 & & 1 & \\ -1 & & & 1 \\ & & & \\ & & & \\ & & & \\ \end{array} \right) \\ \mbox{} % Blank line to match column names so as to align the = vertically \end{array} \\[-12pt] % Correction for blank line \end{align} \]

    この行列は，つぎのことを示している：
    • 写像 \( \partial_1 \) では，６次元のうち３次元が潰れる。
      即ち，\( Ker( \partial_1 ) \) が３次元，\( Ker( \partial_1 ) \) の補空間が３次元。
    • つぎの３つのサイクルが，\( Ker( \partial_1 ) \) の基底を成す： \[ e''_4 = e'_4 + e_1 = ( e_4 - e_2 ) + e_1 = e_1 + e_4 + ( - e_2 ) \\ e''_5 = e'_5 + e_1 = ( e_5 - e_3 ) + e_1 = e_1 + e_5 + ( - e_3 )\\ e''_6 = e'_6 + e_2 = ( e_6 - e_3 ) + e_2 = e_2 + e_6 + ( - e_3 ) \]
    • 有向辺 \( e_1, e_2, e_3 \) が，補空間の基底を成す。